How should I weight the factors that affect the choice of an action in a strategy board game with multiple actions?
You could use a genetic algorithm to optimise the parameter settings. Here you don't choose random parameters all the time, but only at the beginning. Each AI (which is a vector of parameter settings) plays each other one for a ranking (you can probably reduce the number of total games by using a ladder-style ranking where only neighbours play against each other). Also, you don't need 100 to start with, as you might come across better combinations than those present initially throughout the processing.

Then you discard the worst AIs (ie parameter vectors), and recombine the best ones, adding some random mutations into the mix. In theory this should converge faster, as you preserve good parameter settings (depending on how interdependent they are) and remove bad ones from the 'pool'.

You can also have a higher rate of mutations initially, which slowly goes down as you progress through the generations.